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[tex] \lim_{x\to 0}\frac{cos ^2x - cos x^2}{x^2.(e^x + 1)}[/tex]

 

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[laTEX]\lim_{ x\to 0 } \frac{cos^2x-1}{x^2.(e^x+1)} + \lim_{ x\to 0 } \frac{1-cos(x^2)}{x^2.(e^x+1)} = L_1+L_2 \\ \\ L_1: \lim_{ x\to 0 } -\frac{sin^2x}{x^2.(e^x+1)} = - \frac{1}{2} \\ \\ L_2: \lim_{ x\to 0 } \frac{2sin^2(\frac{x^2}{2})}{x^2.(e^x+1)} = \lim_{ x\to 0 } \frac{2sin^2(\frac{x^2}{2})}{\frac{x^2}{2}.2.(e^x+1)} \\ \\ \lim_{ x\to 0 }\frac{sin(\frac{x^2}{2})}{e^x+1} = 0 \\ \\ L = -\frac{1}{2}[/laTEX]

 

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